Frozen Drain Tile

I am working on two different houses that both have downspouts leading into buried tile that is daylighted 20'-30' from the houses. This time of year when the snow on the roof melts, the tiles are, or become frozen and cannot drain the melt. Water then starts backing up into the basements. What are some solutions to clearing the drain tiles? My initial thought would be to fish heat tape through the tiles, but all of the packages that I have seen state that they should not be in direct contact with water.

Use a hydro jetter with a forward piercing nozzle. The running water will melt away the ice. If you do not have access to a hydro jetter you can fish a garden hose down the drain with a nozzle that sprays a hard stream forward. The garden hose idea only works well up to around 50 ' depending if there is any turns in the line or not. One other way is to get one of them waterfall pumps and enough 3/8" hose to run through the line and a 5 gallon bucket full of warm water. Put the pumping the bucket and feed the 3/8" hose down the line. I learned this last idea from an old well guy, he did this to unfreeze the plastic well pump line between the house and the well head.